Arrays of prime ideals in commutative rings

If R is a commutative ring, A and B ideals of R, and S and T multiplicative submonoids of R, we note an elementary necessary and sufficient condition for there to exist prime ideals P and Q in R such that P contains A and is disjoint from S, Q contains B and is disjoint from T, and P Q. We then study conditions for the existence of larger families of prime ideals satisfying similar systems of relations. When the inclusion relations specified in the given system define a “tree order,” the necessary and sufficient conditions are quite tractable; otherwise, they are much less so. We apply these results to the case where R is a tensor product of two algebras over a field k, and end with some observations on the behavior of arrays of prime ideals in a k-algebra under base extension.